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Risk

Terminology

  • Drawdown: Decline from high water mark.

Value-at-Risk (VaR) / Expected Shortfall (ES)

Both are used to measure tail risk. VaR uses \(\alpha\) percentile of the distribution, while ES uses the mean of the tail beyond \(\alpha\) percentile. VaR is easier to compute and understand, but ES is more robust and coherent.

  1. Parameteric: Assume normal distribution and compute VaR/ES using empirical mean and standard deviation. Can be extended to multivariate normal distribution for portfolio VaR/ES.
  2. \(\text{VaR} = -\mu + Z \sigma\), where Z is the upper \(\alpha\) quantile of the standard normal distribution (e.g., \(Z = 1.645\) for \(\alpha = 95\%\)). The negative mean reflects that VaR is expressed as a loss.
  3. \(\text{ES} = \frac{1}{1-\alpha} \int_{\alpha}^{1} VaR_{u} \, du = \sum_{u \ge \alpha} VaR_u\), the average of VaR across all confidence levels \(u \ge \alpha\). Equivalently, for the standard normal with density \(f(x)\): \(ES = \frac{1}{1-\alpha} \int_{-\infty}^{-Z} x \, f(x) \, dx\).
  4. Historical
  5. VaR - sort historical returns and take the \(\alpha\) percentile.
  6. ES - take the mean of the returns below the \(\alpha\) percentile.
  7. EWMA
  8. Compute exponential weights \(w_t = \lambda^{T-t}\) for \(t = 1, \ldots, T\) and \(\lambda \in (0,1)\) is the decay factor. Normalize weights \(\hat{w}_t = \frac{w_t}{\sum_{t=1}^{T} w_t}\) to sum to 1. Equivalently, it is a geometric series where \(\hat{w}_t = \frac{(1-\lambda) \lambda^{T-t}}{1 - \lambda^T}\).
  9. VaR - Sort historical returns based on weights and find smallest index \(k\) such that \(\sum_{t=1}^{k} \hat{w}_t \ge \alpha\). Take the return at index \(k\) as VaR.
  10. ES - Weighted average of returns below the \(\alpha\) percentile, i.e. \(\text{ES}_{1-\alpha} = - \frac{\sum_{t=1}^k \hat{w}_t r_t}{\sum_{t=1}^k \hat{w}_t}\) where \(r_t\) are the returns below the \(\alpha\) percentile.
  11. EWMA Vol Adjusted
  12. Compute EWMA volatility \(\sigma_t = \sqrt{(1-\lambda) r_t^2 + \lambda \sigma_{t-1}^2}\), where \(r_t\) is the return at time \(t\) and \(\lambda \in (0,1)\) is the decay factor. Compute vol-adjusted returns \(\hat{r}_t = \frac{\sigma_T}{\sigma_t} r_t\), where \(\sigma_T\) is the most recent volatility. \(\sigma_1^2 = r_1^2\). Given equal weights \(w_t = \frac{1}{T}\) to all vol-adjusted returns.
  13. VaR - Sort vol-adjusted returns and find smallest index \(k\) such that \(\sum_{t=1}^{k} w_t \ge \alpha\). Take the vol-adjusted return at index \(k\) as VaR.
  14. ES - Weighted average of vol-adjusted returns below the \(\alpha\) percentile, i.e. \(\text{ES}_{1 - \alpha} = - \frac{\sum_{t=1}^k w_t \hat{r}_t}{\sum_{t=1}^k w_t}\) where \(\hat{r}_t\) are the vol-adjusted returns below the \(\alpha\) percentile.